As is usual in physics, one is more interested in local aspects of transformations. As noted earlier, the \(1+1\) D CC transformations, not necessarily globally definedFor a transformation to be globally defined, we shall also demand globally non-singular behavior of the corresponding generators [1]}. In this sense, the other super-translations not included in (REF ) fail to be globally defined. on \(\mathbb {R}\times S^1\) , have the following finite form:
\(x\rightarrow x^\prime =F(x)\hspace{5.69054pt},\hspace{5.69054pt}t\rightarrow t^\prime =tF^\prime (x)+G(x)\) 
